271
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary. This
is a pot like surface [Fig. 8.1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above. Another such surface is shaped like a tiffin box
(without the lid) [Fig. 8.1(c)]. On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq. (8.1) has not changed but the right hand side is zero and not
µ
0
i,
since no current passes through the surface of Fig. 8.1(b) and (c). So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero.
Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something. The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used.
We can actually guess the missing term by looking carefully at
Fig. 8.1(c). Is there anything passing through the surface S between the
plates of the capacitor? Yes, of course, the electric field! If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/ε
0
(see Eq. 2.41). The field is
perpendicular to the surface S of Fig. 8.1(c). It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it. So what
is the electric flux
Φ
E
through the surface S ? Using Gauss’s law, it is
(8.3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq. (8.3), we have
d
d
d
d
d
d
Φ
E
t t
Q Q
t
=
=
ε ε
0 0
1
This implies that for consistency,
= i (8.4)
This is the missing term in Ampere’s circuital law. If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is ε
0
times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces. If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law. B at the point P
is non-zero no matter which surface is used for calculating it. B at a
point P outside the plates [Fig. 8.1(a)] is the same as at a point M just
inside, as it should be. The current carried by conductors due to flow of
charges is called conduction current. The current, given by Eq. (8.4), is a
new term, and is due to changing electric field (or electric displacement,
an old term still used sometimes). It is, therefore, called displacement
current or Maxwell’s displacement current. Figure 8.2 shows the electric
and magnetic fields inside the parallel plate capacitor discussed above.
The generalisation made by Maxwell then is the following. The source
of a magnetic field is not just the conduction electric current due to flowing
FIGURE 8.1 A
parallel plate
capacitor C, as part of
a circuit through
which a time
dependent current
i (t) flows, (a) a loop of
radius r, to determine
magnetic field at a
point P on the loop;
(b) a pot-shaped
surface passing
through the interior
between the capacitor
plates with the loop
shown in (a) as its
rim; (c) a tiffin-
shaped surface with
the circular loop as
its rim and a flat
circular bottom S
between the capacitor
plates. The arrows
show uniform electric
field between the
capacitor plates.